Design FIR Filter with Signed Power of Two Terms Using MATLAB

Design FIR Filter with Signed Power of Two Terms Using MATLAB

Sangita Sol ank

Indore Institute science of Tec hnology, Indore

Abstract: The design of digital filter, using signed power of two ter ms are imple me nte d. The filter are designed without multi pliers. The design is base d on the re mez e xchange algorithm for the design of l ow pass and filters. Line ar phase FIR filter wi th coe fficients constisting of mini mum number of signed power of two ter ms is for mulate d. The ai m of this thesis is first to re duce the region that contains the optimal solution in or der to decre ase the computation ti me and ,then FIR filter are designed using mixe d integer linear pr ogrammi ng. The response obtaine d by this techni que is compare d wi th the simply rounding techni que.

Keyword: re me z e xchange algorith m, Linear phase FIR filter , M ixed integer linear progra mming, Signed power of two terms, Rounding technique.

1 INTRODUCTION

Recently numberous algorith m have been proposed for designing mu ltiplierless finite impulse response (FIR) filters. In mult iplierless digital filter mu ltip licat ion are replaced with a sequence of shift and adds. Therefore only adders are required for the coeffic ient imp le mentation. This leads to significant reduction in the computational comple xity and power consumption.filters with such specifications can be designed by appropriate modification of the McCle llan-Pa rk a lgorith m. By suitable choice of the weighting function of the equirriple error.

The purpose of this paper is to advance a new technique for the design of linear phase FIR filters with equirriple stop band and with a prescribed degree of flatness in the pass band.the proposed technique is based on McClellan-Pa rk algorithm for FIR filter design and optimization is involved here. In Section II the method is introduced along with numerical e xa mp les. For the design of narrow

passband filters, a number of imp roved method are described in Section III, and IV based on low pass and high pass filters. In Section V we discuss certain imple mentation considerations.

II SATEM ENT OF THE PROBLEM

The Parks-McCle llan algorith m, published by James mc Cle llan and Thomas Parks in 1972, is an iterative algorith m for finding the optimal Chebyshev finite impulse responses(FIR) filter.. The Pa rks-Mc Cle llan algorith m is utilized to design and imp le ment efficient and optimal FIR filters. It uses an indirect method for finding the optimal filter coeffic ients.

The goal of the algorith m is to min imize the error in the pass and stop bands by utilizing the Chebyshev approximation. The Parks -McCle llan algorith m is a variation of the Reme z algorith m or Re me z e xchange algorith m, with the change that it is specifica lly designed for FIR filters and has become a standard method for FIR filter design.

Pass and Stop Bands of Parks -Mc Clellan Algorithm

The y-axis is the frequency response H() and the x-

a xis are the various radian frequencies, i. It can be noted that the two frequences marked on the x-a xis, p and s. p indicates the pass band cutoff frequency and s indicates the stop band cutoff frequency. The ripple like plot on the upper left is the pass band ripple and the ripple on the bottom right is the stop band ripple. The two dashed lines on the top left of the graph indicate the p and the two dashed lines on the bottom right indicate the s. All other frequencies listed indicate the ext re mal frequencies of the frequency response plot. As a result there are six e xtre ma l frequencies, and then we add the pass band and stop band frequencies to give a total of eight e xtre ma l frequencies on the plot.

According to the IEEE Signal Processing Magazine,

1. the Parks-McCle llan Algorith m is imple mented using the following steps:

   1. Initia lizat ion: Choose an extre ma l set of frequences {i(0)}.

   2. Fin ite Set Approximat ion: Ca lculate the best

      Chebyshev approximation on the present e xtre ma l set, giving a value (m) for the min- ma x e rror on the present extre mal set.

   3. Interpolation: Ca lculate the error function E() over the entire set of frequencies using (2).

   4. Look for loca l ma xima of |E(m)()| on the set .

      i

   5. If ma x()|E(m)()| > (m), then update the e xtre ma l set to { (m+1)} by picking new frequencies where |E(m)()| has its local ma xima. Make sure that the error alternates on the ordered set of frequencies as described in (4) and (5). Return to Step 2 and iterate.

   6. If ma x()|E(m)()| (m), then the algorithm is complete. Use the set {i(0)} and the interpolation formu la to co mpute an inverse discrete Fourier transform to obtain the filter coeffic ients.

According the Professor Douglas Jones of the University of Illinois, [4]the Pa rks-Mc Cle llan Algorith m may be imp le mented as the following:

1. Make an initia l guess of the L+2 e xtre ma l frequencies.

2. Co mpute using the equation given.

1. Using Lagrange Interpolation, we co mpute the dense set of samples of A() over the passband and stopband.

2. We Determine the new L+2 largest e xtre ma .

3. If the Alternation Theore m is not satisfied, then we go back to (2) and iterate until the Alternation Theorem is satisfied.

4 If the Alternation Theorem is satisfied, then we Co mpute h(n) and we a re done.

To gain a basic understanding of the Parks -Mc Cle llan Algorith m mentioned above, we can rewrite the algorith m above in a simp ler form as:

1. Guess the positions of the extre ma are evenly spaced in the pass and stop band.

2. Perform polynomia l interpolation and re- estimate positions of the local e xtre ma.

3. Move extre ma to new positions and iterate until the e xtre ma stop shifting.

III PROBLEM ANA YSIS

Exa mple 1: Design and plot the equirriple linear phase FIR low pass filter with order 36.using signed power of two terms. Normalized frequency of pass band and stop band are 0.15, 0.25. also design this filter using rounding method.

TABLE 1

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Cofficients of H(z ) in Example 1

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Filter length =36

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h(0) = -0.0088 = h(36)

h(1) = -0.0069 = h (35)

h(4) = 0.0074 = h(32) h(5) = 0.0131 = h(31) h(6) = 0.0162 = h(30) h(7 ) = 0.0111 = h(29) h(8) = 0.0003 = h(28) h(9) = -0.0166 = h(27)

h(10) = -0.0307 = h(26)

h(11) = -0.0375 = h (25)

h(12) = -0.0275 = h (24)

h(13) = -0.0002 = h(23) h(14) = 0.0449 = h(22) h(15) = 0.0979 = h(21) h(16) = 0.1498 = h(20) h(17) = 0.1861 = h(19) h(18) = 0.2002

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IV IMPLEM ENTATION CONSIDERATION

FIR dig ital filter designed over the signed power of two (SPT) discrete spaces were first proposed by Lim and Constantinides. This section briefly describes the SPT nu mber characteristics and exiting optimization techniques for the design of digital filter subject to SPT coeffic ient.

1. Signed Powe r of Two

   In mathe matics, a powe r of two is any of the ineger powers of the number t wo.because two is the base of the binary system, powe r of t wo are important to computer science.

2. Signed Digit Representation

The radix-2 signed digit format is a 3-valued representation of a radix-2 nu mber and emp loys three digit values 0,1 and -1.A simp le a lgorith m

representation which convert radix-2 binary nu mber to equivalent SD representation is as follow.

C i = a i-1 – a i , i = b ,

b-1, ..1,

After that decima l nu mber is represent the binary number using this formula

TAB LE II

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Imple me nte d Result

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Cofficients of H(z) in Example 1 leng th =36

———————————————————–

h(0) = 0 = h(36)

h(1) = 0 = h(35)

h(2) = 0 = h(34)

h(3) = 0 = h(33)

h(4) = 0 = h(32)

h(5) = 0 = h(31)

h(6) = 0 = h(30)

h(7) = 0 = h(29)

h(8) = 0 = h(28)

h(9) = 0 = h(27)

h(10) = 0 = h(26) h(11) = -2-5 = h(25)

h(12) = 0 = h(24)

h(13) = 0 = h(23) h(14) = -2-5 = h(22)

h(15) = 2-4+2-5 = h(21)

h(16) = 2-3 = h(20)

h(17) = 2-3+2-5 = h(19)

h(18) = 2-3+2-5

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Fig. 1 Frequency response optimized with length N= 36 obtained , rounding the coefficient value

Fig. 2 Frequency response optimized with length N= 36 obtained , rounding the coefficient value

V CONCLUSION

In this paper the methods of integer linear programming are particularly useful for designing FIR filters with the power of two coefficients. The result obtained is significant when co mpared to simp le rounding of coeffic ient value.the aim of optimization is only the minimizat ion of the number of SPT terms.Extensive research has shown that the comple xity of an FIR filter can be reduced by imple mentation its coeffic ient as sum of SPT terms and faster hardware imple mentation of the mu ltip licat ion operation.

REFRRENCES

1. J.H.Mc Cle llan, T.W.Park and L.R.Rab iner, FIR linear phase filter design progra m, in progra m for dig ital signal processing, New York: IEEE Press,1979, pp.5.1.1-5.1.13.

2. Y.C.Lim, and S.R.Parker, FIR filter design over a discrete power t wo coeffic ient space,IEEE Trans. Acoust. Speech, signal processing ling, Vo l. ASSP-31, June 1983.

3. Y.C.Lim, and S.R.Parker, and A.G.Constanntinides, Finite word length FIR filter design using integer programming over a discrete coeffic ient space,IEEE Trans. Acoust. Speech, signal processing ling, Vo l. ASSP-30,pp. 661-664, Agu. 1982.

4. Mit ra, Kaiser, Handbook for digita l processing, john wiley &sons, 1993.

5. http://www.ie.ncsu.edu/kay/matlab

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